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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> Fredholm determinant in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="noindex,nofollow" /> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } </style> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script type="text/javascript"> <![CDATA[//><!]]> </script> </head> <body> <div id="Container"> <textarea id='content' readonly=' readonly' rows='24' cols='60' >_Fredholm determinant_ is a generalization of a determinant of a finite-dimensional matrix to a class of operators on Banach spaces which differ from identity by a trace class operator or by an appropriate analogue in more abstract context (there are appropriate determinants on certain Banach ideals). It is often considered as an analytic function of a perturbation parameter $\lambda$. The calculations with Fredholm determinants have applications in operator theory, [[random matrix]] theory, integrable models etc. In his original setup, Fredholm attached the determinant $$ d(\lambda) = \sum_{n=0}^{\infty} \frac{\lambda^n}{n!}\int_a^b\int_a^b\ldots \int_a^b det((K(x_k, x_l))_{k,l=1}^n) d x_1 \cdots d x_n $$ to the Fredholm integral equation of the second kind $$ u(x) + \lambda \int^b_a K(x,y) u(y) d y = f(x), \,\,\,x\in (a,b), $$ and $d$ is an [[entire function]] of $\lambda$ such that $d(\lambda) \neq 0$ iff the integral equation has a unique solution. ## References #### General and mathematical * wikipedia [Fredholm determinant](http://en.wikipedia.org/wiki/Fredholm_determinant) * Albrecht Pietsch, _Operator ideals_, North-Holland Mathematical Library 20; _Eigenvalues and s-Numbers_, Cambridge Studies in Advanced Mathematics 13; _History of Banach spaces and linear operators_, sec 6.5.2 * [[Alexander Grothendieck]], _La th&#233;orie de Fredholm_, Bulletin de la Soci&#233;t&#233; Math&#233;matique de France __84__ (1956), p. 319-384, [numdam](http://www.numdam.org/item?id=BSMF_1956__84__319_0) * Fritz Gesztesy, Yuri Latushkin, Konstantin A. Makarov, _Evans functions, Jost functions and Fredholm determinants, Arch. Rational Mech. Anal. 186 (2007) 361&#8211;421,[doi](http://dx.doi.org/10.1007/s00205-007-0071-7) [pdf](http://www.math.missouri.edu/~yuri/preprints/EJF.pdf) * Barry Simon, _Notes on infinite determinants of Hilbert space operators_, Advances in Math. __24__ (1977), no. 3, 244&#8211;273 [MR482328](http://www.ams.org/mathscinet-getitem?mr=482328) <a href="http://dx.doi.org/10.1016/0001-8708(77)90057-3">doi</doi> * Folkmar Bornemann, _On the numerical evaluation of Fredholm determinants_, Math. Comp. 79 (2010), no. 270, 871&#8211;915 [MR2600548](http://www.ams.org/mathscinet-getitem?mr=2600548) [pdf](http://www-m3.ma.tum.de/foswiki/pub/M3/Allgemeines/FolkmarBornemannPublications/Fredholm.pdf) [doi](http://dx.doi.org/10.1090/S0025-5718-09-02280-7) #### Physics applications * F. J. Dyson, _Fredholm determinants and inverse scattering problems_, Comm. Math. Phys. 47, 171&#8211;183 (1976) [MR406201](http://www.ams.org/mathscinet-getitem?mr=406201) [euclid](http://projecteuclid.org/euclid.cmp/1103899727) * C. A. Tracy, H. Widom, _Fredholm determinants, differential equations and matrix models_, Commun. Math. Phys. __163__, 33&#8211;72 (1994) [MR1277933](http://www.ams.org/mathscinet-getitem?mr=1277933) [euclid](http://projecteuclid.org/getRecord?id=euclid.cmp/1104270379) * J. Harnad, Alexander R. Its, _Integrable Fredholm operators and dual isomonodromic deformations_, Comm. Math. Phys. 226 (2002), no. 3, 497&#8211;530 [MR1896879](http://www.ams.org/mathscinet-getitem?mr=1896879) [doi](http://dx.doi.org/10.1007/s002200200614) * M. Adler, [[M. Cafasso]], P. van Moerbeke, _Nonlinear PDEs for Fredholm determinants arising from string equations_, [arxiv/1207.6341](http://arxiv.org/abs/1207.6341) * M. Bertola, [[M. Cafasso]], _Fredholm determinants and pole-free solutions to the noncommutative Painlev&#233; II equation_, Comm. Math. Phys. __309__ (2012), no. 3, 793&#8211;833 [MR2885610](http://www.ams.org/mathscinet-getitem?mr=2885610) [doi](http://dx.doi.org/10.1007/s00220-011-1383-x) * Michio Jimbo, Tetsuji Miwa, Yasuko M&#244;ri, Mikio Sato, _Density matrix of an impenetrable Bose gas and the fifth Painlev&#233; transcendent_, Phys. D 1 (1980), no. 1, 80&#8211;158 [MR573370](http://www.ams.org/mathscinet-getitem?mr=573370) <a href="http://dx.doi.org/10.1016/0167-2789(8090006-8)"> category: functional analysis [[!redirects Fredholm determinants]]</textarea> </div>  </body> </html>